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105x a

+b=-*; .. 100

100 a translation of the enunciation, by supposing the interest additive to the capital, in which case, the sum b ought to be subtracted, This equation, treated as the preceding, shall give

x=206, If the interest at 5 per cent be subtracted from 100, in which case 100 reduces itself to 95, we have the capital c at the end of the year, by the proportion

100: 95 : ;3 : = the capital;


95x consequently,


100 multiplying by 100, and transposing, we shall have

100b=5x, s. x=206. The negative isolated result, that is, the negative value of x, would announce a rectification or a correction in the terms of the enunciation, and the problem proposed could be re-established in two ways.


Prob 5. What number is that, the double of which exceeds its half by 6?


Let x=the number;
Then by the problem, 2x= :6,

.. multiplying by 2, 42% 12,

or 3x=12, .. by division, x=4.

Prob. 6. From two towns which are 187 miles distant, two travellers set out at the same time, with an intention of meeting. One of them goes 8 miles, and the other 9 miles a day. In how many days will they meet ?

Let x= the number of days required; then 8x=the number of miles one travelled, and 9x = the number the other travelled ;

and since they meet, they must have travelled together the whole distance,

consequently, 8x+9x=187,

or 17x=187, .. by division, x=11.

Prob. 7. What number is that, from which 6 being subtracted, and the remainder multiplied by 11, the product will be 121 ?

Let w=the number required ; Then by the problem (-6) XH=121, by transposition, 112=121 +66,

or 11x=187, .. by division, a=17.

Prob. 8. A Gentleman meeting 4 poor persons, distributed five

4 shillings amongst them: to 1' second he gave twice, the third thrice, and to the fourth four times as much as to the first. What, did he give to each ?

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Let <= the pence he gave to the first,

..2x= the pence given to the second, and 3x =

to the third, 4x

to the fourth .. by the problem, uto2x+3x+4x=5X12=60,

or 10x=60,

by division, x=6, and therefore he gave 6, 12, 18, 24 pence respectively to them.

Prob. 9. A Bookseller sold 10 books at a certain price; and afterwards 15 more at the same rate. Now at the latter time he received 35 shillings more than at the former. What did he receive for each book ?

Let = the price of a book.

then 10x= the price of the first set, and 15x=the price of the second set ; bat by the problem, 151=10x +35;

.. by transposition, 5x=35;

and by division, x=7,

Prob. 10. A Gentleman dying bequeathed a legacy of 1400 dallars to three servants. A was to have twice as much as B; and B three times as much as C. What were their respective shares ?

Let x=C's share,

.. 3x=B's share,

and 6x=A's share ; then by the problem, x+3x+6x=1400,

or 10x = 1400,

.. by division, x=140=C's share. ... A received 840 dollars; B, 420 dollars; and C, 140 dollars.

Prob. 11. There are two numbers whose difference is 15, and their sum 59. What are the numbers ?

As their difference is 15, it is evident that the greater number must exceed the lesser by 15. Let, therefore, x= the lesser number; then will +15= the greater; ... by the problem, x++15=59,

or 2.+15=59, by transposition, 2x=59-15=44, ... by division, x=22 the lesser number,

and x+15=22 +15=37 the greater.

Prob. 12. What two numbers are those whose difference is 9; and if three times the greater be added to five times the lesser, the sum shall be 35 ?

Let the lesser number;
then *+9= the greater number.
And 3 times the greater=3(x+9)=3x+27,
5 times the lesser =5x.

.. by the problem, (3x+27)+5x=35
by transposition, 3x+5x=35-27,

or 8x=8; .. by division, x=1 the lesser number, and x+9=1+9=10 the greater number.


Prob. 13. What number is that, to which 10 being added, şths of ihe sum will be 66 ?

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Let x= the number required; then x+10= the number, with 10 added to it.

3(x+10) Now ths of (~ +10)=3(x+10)=3x +30

But, by the problem, zths of (x +10)=66;

3x +30



by multiplication, 3.x +30=330;

by transposition, 3x=300;

.. by division, x=100.

Prob. 14. What number is that, which being multiplied by 6, the product increased by 18, and that sum divided by 9, the quotient shall be 20?


Let x= the number required , then 6x= the number multiplied by 6;

6x +18= the product increased by 18; and 6x +18

= that sum divided by 9. 9

6x +18 .. by the problem,

= 20;

9 by multiplication, 6x +18=20X9; by transposition, 6x=180-18,

or 6x=162; .. by division, =27.

Prob. 15. A post is {th in the earth, ths in the water, and 13 feet out of the water. What is the length of the post ?

Let x= the length of the post;

2 then the part of it it in the earth,



= the part of it in the water,

7 and 13= the part of it out of the water.

But by the problem, part in earth + part in water + part out of water = whole part;

()+(**) +13=«.

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30 and X35+ X 35+13 X 35=35x,


or 70 +15x+455=35x; by transposition, 455=351---7-15x=13x,

or 13x = 455 ; ... by division, x=35 length of the post.

Prob. 16. After paying away {th and th of my money, I had 850 dollars left. What money had I at first ?

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x= the


my purse at first; then money paid away.

4 7 But money

at first money paid away = money remaining .. by the problem,



or a


4 7 Multiplying by 28 the product of 4 and 7, which is the least common multiple, and 28x - *28-X 28=850 28,

or 28. — 7x 4x=23800, ... 17:=23800 ; and by division, x=1400 dollars.

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Prob. 17. What number is that, whose one half and one third plus 12, shall be equal to itself?

fhen, by the problem, r= + +12;

Let = the number required ; , = -+



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